The \(\epsilon \) -dependent compensating diffeomorphisms and reshufflings between \(\chi \)  and \(\mu \)  generated by \( \xi ^i(\epsilon ) \mathcal {H}_i + \lambda ^T(\epsilon ) \mathcal {F}\)  are included to preserve the boundary conditions (specifically \(\bar{h}_{rA} = 0\)  and \(\mu _i = \mathcal {O}(r^{-2})\) , where \(\bar{h}_{rA}\)  is the radial-angular component of the leading order of the metric [1]}, as well as the absence of zero-mode condition for \(\chi ^{(0)}\)  and \(\omega \) ). One has (to leading \(\mathcal {O}(1)\)  order)
\(\xi ^i (\epsilon ) = -\frac{i}{2}\partial ^{i}\left(r\epsilon ^{T}\gamma _{0}\gamma _{r}\right)\chi \,, \quad \lambda (\epsilon ) = -\big (\epsilon - \epsilon _{0}\big )\,,\) 
